Below separation a cylinder's drag is a Stokes layer, and KC stops deciding the force

Long enough to make a wake put the Keulegan–Carpenter number at the centre of Morison’s equation. An oscillating flow of amplitude $U_0$ and period $T$ past a cylinder of diameter $D$ exerts a force that Morison wrote as an inertia term, proportional to the acceleration, plus a drag term, quadratic in the velocity; their peaks are equal when $\mathrm{KC} = U_0T/D$ reaches $\pi^2 C_M/C_D$, about 16.45 for typical coefficients, and below that the inertia term is the force. Two forces, and only one of them remembers read the two terms as two kinds of memory — the inertia term none, the drag term the wake shed in the previous half cycle — and found the drag’s share running from two per cent to eighty-six across the range a designer meets.

Both essays had a wake in them, and both named a second number that would not go away: the frequency parameter $\beta = D^2/(\nu T)$, the Reynolds number divided by KC. Below a KC of a few, the flow does not separate at all. The fluid oscillates round the cylinder in its potential flow, with a thin boundary layer on the surface that has no time to grow before the motion reverses. There is no wake to remember, and the drag term is describing something else.

What it describes is the Stokes layer, and its drag coefficient grows as one over KC — which exactly cancels the Keulegan–Carpenter number’s say in the split. Below separation the drag’s share of the force is a function of $\beta$ alone. The number that decided everything above separation decides nothing below it.

A layer that cannot grow #

At small KC the outer flow is the potential flow of an oscillating stream past a circle: the fluid slips along the surface at twice the stream’s speed times the sine of the angle from the front, reversing every half period. Underneath that slip the no-slip condition makes a boundary layer, and since the motion reverses before a steady layer could form, the layer is the oscillating one the wall that shakes computed for a plate: a profile whose thickness is $\sqrt{2\nu/\omega}$, fixed by the viscosity and the frequency rather than by any distance run. For a cylinder that thickness, over the diameter, is of order $\beta^{-1/2}$, which is why $\beta$, and not the Reynolds number, is the number that measures it.

Wang solved the problem in 1968 by matched asymptotic expansions, carrying the inner layer and the outer potential flow to three terms in $(\pi\beta)^{-1/2}$. In Morison’s language his result is two coefficients:

$$C_D = \frac{3\pi^3}{2\,\mathrm{KC}}\left[(\pi\beta)^{-1/2} + (\pi\beta)^{-1} - \tfrac14(\pi\beta)^{-3/2}\right],\qquad C_M = 2 + 4(\pi\beta)^{-1/2} + (\pi\beta)^{-3/2}.$$

The drag coefficient is defined as Morison’s is: the coefficient of the quadratic drag term that dissipates the same energy over a cycle as the real force. The inertia coefficient’s first term, two, is the potential flow’s — the displaced fluid plus the added mass of the mass a body has to borrow — and the rest is the Stokes layer thickening the body the outer flow sees.

A drag coefficient that grows without limit #

The first figure draws Wang’s drag coefficient against KC for four values of $\beta$, each curve stopping at the edge of the regime. At $\mathrm{KC} = 1$ it is 2.77 for $\beta = 10^2$, 0.845 for $10^3$, 0.264 for $10^4$ and 0.083 for $10^5$, falling as the square root of $\beta$ as the layer thins. And at fixed $\beta$ it rises as one over KC, without limit: halve the motion’s amplitude and the drag coefficient doubles.

That divergence is not a breakdown. It is what a drag force linear in the velocity looks like when it is written in a quadratic form. The Stokes layer’s shear stress is proportional to the slip velocity, so the real drag is linear in $U_0$; Morison’s term is quadratic, $\tfrac12\rho D C_D|U|U$; matching the two forces $C_D$ to carry a factor $1/U_0$, and with the period and diameter fixed that is $1/\mathrm{KC}$. A drag coefficient that diverges at small KC is the signature of a viscous drag being forced into an inertial form, the same signature the world with no inertia showed for a sphere’s drag coefficient at low Reynolds number.

The share that KC no longer decides #

The consequence is the second figure. The ratio of the drag term’s peak to the inertia term’s is $C_D\,\mathrm{KC}/(\pi^2 C_M)$, the line the earlier essay found to nine figures. Above separation, with $C_D$ and $C_M$ roughly constant, it grows in proportion to KC, and that is why KC decided the split. Below separation $C_D\,\mathrm{KC}$ does not contain KC at all, and the ratio is flat: 0.126 for $\beta = 10^2$, 0.041 for $10^3$, 0.013 for $10^4$ and 0.0042 for $10^5$, at every KC up to the onset.

So in the attached regime the drag’s share is set entirely by how thick the Stokes layer is compared with the cylinder, and it is small — a few per cent at most for any $\beta$ above a hundred. The force is the inertia term, and the drag term’s only job is the energy: it is the whole of the cycle’s dissipation, since the inertia term, as the first essay found, does no work at all. A designer who estimates a small-amplitude force from Morison’s equation with a separated flow’s drag coefficient will get the force about right, because inertia dominates either way, and the damping wrong by whatever factor the two drag coefficients differ by.

What the layer adds to the inertia #

The third figure draws the inertia coefficient. It is 2.226 at $\beta = 10^2$, 2.071 at $10^3$ and 2.007 at $10^5$, approaching the potential flow’s two as the layer thins. The excess is the Stokes layer’s displacement: a layer of thickness $\delta$ makes the cylinder look larger to the outer flow, and its oscillation is out of phase with the stream’s, so part of its effect appears as extra inertia and part as drag. The two corrections are the same physics divided between the two terms by their phase, which is why $C_M - 2$ and $C_D\,\mathrm{KC}$ are both proportional to $\beta^{-1/2}$ at leading order.

The truncation is invisible. The dashed two-term curve and the three-term one coincide to a hundredth of a per cent even at $\beta = 10^2$, where the second term itself is a tenth of the whole; the expansion converges fast wherever the layer is thinner than about a tenth of the radius, which every $\beta$ drawn here satisfies.

Where the attached flow ends #

The fourth figure is the regime’s boundary. Before the flow separates in two dimensions, the attached oscillating flow becomes unstable in three: Honji found in 1981 that the Stokes layer breaks into rows of mushroom-shaped vortices along the cylinder’s span, and Hall’s linear stability analysis of 1984 put the onset at $\mathrm{KC} \approx 5.78\beta^{-1/4}$ for large $\beta$. Wang’s coefficients hold below that line.

The line falls as $\beta$ grows: the attached regime ends near KC 1.83 at $\beta = 10^2$, 1.03 at $10^3$, 0.58 at $10^4$ and 0.33 at $10^5$. In Reynolds numbers the onset rises, from about a thousand at $\beta = 10^3$ to about thirty thousand at $10^5$, so the regime is not a low-Reynolds-number one; it is a small-excursion one. A thick cylinder in a fast oscillation is attached only if the fluid moves a small fraction of its diameter.

The separated crossover, KC = 16.45, sits far above every point of the line. Between the two lies the region the earlier essays worked in, where the flow separates, the wake forms and is swept back, and the coefficients are measured rather than derived. The attached regime is the corner of the map where they can be derived — and where they say something the measured ones never needed to, that KC has stopped mattering.

The drag is the layer’s dissipation #

The fifth figure checks the leading term by a route that shares nothing with Wang’s expansion. The Stokes layer under a slip velocity oscillating at amplitude $U_s$ has a known profile, and the energy it dissipates per unit area is the mean of $\mu(\partial u/\partial y)^2$ integrated through it; integrating that profile numerically gives $\rho U_s^2\sqrt{\nu\omega}/(2\sqrt2)$ to one part in a hundred thousand. Round the cylinder the slip is $2U_0\sin\theta$, so the total dissipation per unit length is that density integrated over the perimeter. Equating it to the mean power of Morison’s drag term, $\tfrac12\rho D C_D U_0^3 \cdot 4/(3\pi)$, gives a drag coefficient — and it equals Wang’s leading term to a part in $10^5$ at every $\beta$.

The agreement is the clearest statement of what the drag coefficient means here. It is not a measure of a wake’s momentum, as it is above separation; it is the Stokes layer’s dissipation, written as though it were. Wang’s second and third terms, which the energy route does not contain, add 5.6 per cent at $\beta = 10^2$ and 0.56 per cent at $10^4$: they are the curvature of the cylinder and the layer’s interaction with the outer flow, which the flat-plate profile ignores.

The phase says which regime a measurement is in #

A measured force record cannot be split into drag and inertia by its size alone; it is split by its phase. Morison’s inertia term is in phase with the acceleration and his drag term with the velocity, a quarter cycle apart, and the earlier essay’s point was that a peak-force measurement throws away that phase. Below separation the phase is the one quantity that identifies the regime from the record. The drag share is a few per cent, so the total force leads the velocity by nearly a quarter cycle — by $\arctan$ of the inertia peak over the drag peak, about 89° at $\beta = 10^4$ — and the angle is the same at every amplitude. Above separation the drag share grows with KC and the force’s phase swings towards the velocity as the amplitude rises. A record whose phase does not change with the amplitude is attached; one whose phase moves is shedding a wake.

That is a usable test because both quantities are measured routinely. A force balance and a velocity probe give the phase directly, and a sweep over amplitude at one frequency — one $\beta$ — either holds the phase still or does not. Where it holds still, Wang’s coefficients apply and the damping can be computed rather than measured; where it starts to move, Hall’s onset has been passed.

Why a quadratic coefficient is the wrong currency here #

Morison’s equation was written for waves on piles, where the flow separates and the drag really is quadratic: a wake’s momentum deficit grows with the square of the velocity that makes it. Its coefficients are then numbers of order one that vary slowly, and tabulating them against KC and $\beta$ is natural. In the attached regime the same tabulation becomes awkward: the drag coefficient runs to tens at small KC, and every table entry carries a hidden $1/\mathrm{KC}$. The honest quantity is the linear damping — the force per unit velocity, $C_D\,\mathrm{KC}$ times a constant — which is flat in KC and falls as $\beta^{-1/2}$.

The pattern is a general one, and it recurs across fluid mechanics. The Reynolds number, and the length in it is the warning that a dimensionless coefficient inherits whatever its definition divided by. A drag coefficient divides a force by a dynamic pressure, which is the right scale only when the force is inertial. Applied to a viscous force it manufactures a dependence on the very number it was meant to collapse away.

Where the attached regime is met #

In the sea the regime is small. A pile half a metre across in a ten-second swell has $\beta$ of 25,000, and Hall’s onset at KC 0.46, which is an excursion of under four centimetres: real waves separate the flow round any structure that size. Large structures meet small KC for a different reason — their diameter is a fraction of the wavelength and the force is diffraction, not Morison’s — and small ones meet large KC.

The regime belongs instead to small things moving quickly, and to things vibrating in still fluid, which is the same problem in a moving frame. A heat exchanger’s tubes vibrating in their coolant, a vibrating-tube densitometer, a tuning-fork viscometer and the prongs of a quartz sensor all oscillate through excursions far smaller than their size, and their damping is exactly this Stokes-layer drag. What a fluid takes out of a swing computed it for a pendulum; a viscometer measures $\beta$ by measuring the damping, which is the drag coefficient times KC that this essay found independent of the amplitude. For a cylinder vibrating in still fluid the inertia coefficient’s two becomes one — the displaced fluid is not accelerated by a pressure gradient when the fluid at infinity is still — but the Stokes layer’s drag and its addition to the inertia are the same.

The steady streaming the oscillation drives, the second-order flow that an oscillation with somewhere to go found, is also part of this regime, and at the larger KC near the onset it is the streaming’s cells, not the first-order layer, that become unstable.

The convention: Morison’s coefficients, energy-matched #

The flow is $U_0\cos\omega t$ past a fixed cylinder; KC is $U_0T/D$, $\beta$ is $D^2/(\nu T)$, and the Reynolds number $U_0D/\nu$ is their product. The drag coefficient is the one whose quadratic Morison term dissipates as much energy over a cycle as the real force — the Fourier-averaged definition, not a peak fit — and the inertia coefficient is the in-phase force over $\rho\pi D^2/4$ times the acceleration. Hall’s onset is quoted at its leading order in $\beta^{-1/4}$; his higher-order corrections raise it slightly at the smaller $\beta$.

What the expansion leaves out #

Separation. The coefficients hold only while the flow is attached and two-dimensional, below Hall’s onset; above it the Honji vortices add a drag of their own, and above KC of a few the flow separates and the earlier essays’ measured coefficients take over.

Roughness and the cylinder’s shape. Wang’s result is for a smooth circular cylinder. Roughness of the order of the Stokes layer’s thickness — micrometres in water at a high frequency — changes the drag at once.

Three-dimensional ends. The cylinder is infinitely long; a finite one’s ends carry their own layers and their own added mass.

Who worked it out #

Stokes computed the oscillating layer on a plate and the damping of a pendulum in 1851; Wang’s matched expansion for the cylinder is from 1968; Honji photographed the three-dimensional instability in 1981, and Hall analysed it in 1984; Sarpkaya’s experiments of 1986 measured the coefficients in this regime and found Wang’s drag holding up to the onset. The observation that below the onset the Keulegan–Carpenter number drops out of Morison’s split is a line of algebra on Wang’s result, drawn out here.

How it was checked #

The Stokes layer’s dissipation, integrated numerically through its profile and averaged over the cycle, matches its closed form to $9\cdot10^{-6}$. The energy route’s drag coefficient matches Wang’s leading term to the same precision at three values of $\beta$. The drag share’s independence of KC is algebraic, and the figure’s flat lines are computed rather than assumed.

Still open: the streaming that becomes the instability #

The second-order streaming an oscillating cylinder drives is a steady flow of four cells, towards the cylinder along the oscillation’s axis and away across it, and being second order in the amplitude it grows as KC squared. Hall’s onset is an instability of the oscillating layer, but near it the streaming is no longer small, and whether the Honji vortices are better described as the oscillating layer’s instability or as the streaming cells breaking into the span is a question the linear analysis does not settle. The next calculation would compute the steady streaming’s strength at Hall’s onset across $\beta$, compare its own Reynolds number with the values at which steady cellular flows become unstable, and ask whether the onset line is where the streaming, rather than the oscillating layer, first loses stability.